Area: Formulas and Derivations This slide show was written by Michael Braverman Trenton, NJ March 2015

Area: Formulas and Derivations A = bh h b All area formulas are based on the formula of a rectangle: Area of a rectangle = base x height.

Area: Why do we use square units?

Area: Why do we use square units?

Area: Why do we use square units? This has an area of 12 squares, one unit on each side. These are called unit squares and have an area of 1 square unit (1 u2) each.

Area: How do we show work? 1. Begin with a General Formula A General Formula is a rule that uses variables to represent parts that will always be there. 3 units 4 units

Area: How do we show work? 1. Begin with a General Formula The General Formula (when applied correctly) ALWAYS gives the correct answer. 3 units 4 units

Area: How do we show work? 1. Begin with a General Formula Area of a Rectangle = base x height 3 units 4 units

Area: How do we show work? 1. Begin with a General Formula Area of a Rectangle = base x height Note: base and height are better terms to use than length and width because of the way the other formulas work. 3 units 4 units

Area: How do we show work? 1. Begin with a General Formula Area of a Rectangle = base * height Abbreviate this as: A = b * h 3 units 4 units

Area: How do we show work? 2. Substitute in the information that you know. A = b * h A = 4u * 3u NOTE: Keep your units throughout your work to prevent careless errors! 3u 4u

Area: How do we show work? 3. Simplify. Be sure to write your units correctly! A = b * h A = 4u * 3u A = 12u2 3u 4u

Area of a Parallelogram A = bh 5 units 4 units 8 units

Area of a Parallelogram A = bh 5 units 4 units 8 units

Area of a Parallelogram A = bh 5 units 4 units 8 units

Area of a Parallelogram A = bh 5 units 4 units 8 units

Area of a Parallelogram A = bh 5 units 4 units 8 units

Area of a Parallelogram A = bh 5 units 4 units 8 units

Area of a Parallelogram A = bh 5 units 8 units 4 units

Area of a Parallelogram A = bh 5 units 8 units 4 units

Area of a Parallelogram A = bh 5 units 8 units 4 units

Area of a Parallelogram A = bh 5 units 4 units 8 units

Area of a Parallelogram 4 units 5 units 8 units

Area of a Parallelogram 4 units 5 units 8 units

Area of a Parallelogram A = bh 4 units 5 units 8 units

Area of a Parallelogram A = bh 5 units 4 units 8 units

Area of a Parallelogram A = bh 5 units 4 units 8 units

Area of a Parallelogram A = bh 5 units 4 units 8 units

Area of a Parallelogram A = bh A = 8u * 4u 5 units 4 units 8 units

Area of a Parallelogram A = bh A = 8u * 4u 5 units 4 units A = 32u2 8 units Note that the “5 units” is NOT used in the area at all. (It is part of the original perimeter)

Area of a Triangle All triangles can be “cloned” and “flipped” to form a parallelogram. This means that the area of a triangle is 1/2 the area of a parallelogram.

Area of a Triangle All triangles can be “cloned” and “flipped” to form a parallelogram. This means that the area of a triangle is 1/2 the area of a parallelogram.

Area of a Triangle All triangles can be “cloned” and “flipped” to form a parallelogram. This means that the area of a triangle is 1/2 the area of a parallelogram.

Area of a Triangle All triangles can be “cloned” and “flipped” to form a parallelogram. This means that the area of a triangle is 1/2 the area of a parallelogram.

Area of a Triangle All triangles can be “cloned” and “flipped” to form a parallelogram. This means that the area of a triangle is 1/2 the area of a parallelogram.

Area of a Triangle All triangles can be “cloned” and “flipped” to form a parallelogram. This means that the area of a triangle is 1/2 the area of a parallelogram.

Area of a Triangle All triangles can be “cloned” and “flipped” to form a parallelogram. This means that the area of a triangle is 1/2 the area of a parallelogram.

Area of a Triangle All triangles can be “cloned” and “flipped” to form a parallelogram. This means that the area of a triangle is 1/2 the area of a parallelogram.

Area of a Triangle All triangles can be “cloned” and “flipped” to form a parallelogram. This means that the area of a triangle is 1/2 the area of a parallelogram.

Area of a Triangle All triangles can be “cloned” and “flipped” to form a parallelogram. This means that the area of a triangle is 1/2 the area of a parallelogram.

Area of a Triangle All triangles can be “cloned” and “flipped” to form a parallelogram. This means that the area of a triangle is 1/2 the area of a parallelogram.

Area of a Triangle All triangles can be “cloned” and “flipped” to form a parallelogram. This means that the area of a triangle is 1/2 the area of a parallelogram.

Area of a Triangle All triangles can be “cloned” and “flipped” to form a parallelogram. This means that the area of a triangle is 1/2 the area of a parallelogram.

Area of a Triangle All triangles can be “cloned” and “flipped” to form a parallelogram. This means that the area of a triangle is 1/2 the area of a parallelogram.

Area of a Triangle All triangles can be “cloned” and “flipped” to form a parallelogram. This means that the area of a triangle is 1/2 the area of a parallelogram.

Area of a Triangle This is a copy of the bottom side. All triangles can be “cloned” and “flipped” to form a parallelogram. This means that the area of a triangle is 1/2 the area of a parallelogram.

Area of a Triangle This is a copy of the left side. All triangles can be “cloned” and “flipped” to form a parallelogram. This means that the area of a triangle is 1/2 the area of a parallelogram.

Area of a Triangle This “side” is Congruent to itself. All triangles can be “cloned” and “flipped” to form a parallelogram. This means that the area of a triangle is 1/2 the area of a parallelogram.

Area of a Triangle Similarly: These Angles are congruent. All triangles can be “cloned” and “flipped” to form a parallelogram. This means that the area of a triangle is 1/2 the area of a parallelogram.

Area of a Triangle Similarly: These So are these. Angles are congruent. All triangles can be “cloned” and “flipped” to form a parallelogram. This means that the area of a triangle is 1/2 the area of a parallelogram.

Area of a Triangle Similarly: These And these. Angles are congruent. All triangles can be “cloned” and “flipped” to form a parallelogram. This means that the area of a triangle is 1/2 the area of a parallelogram.

Area of a Triangle Therefore, the area of a triangle = the area of a parallelogram divided by 2.

Area of a Triangle Therefore, the area of a triangle = the area of a parallelogram divided by 2.

Area of a Trapezoid Base 1 height Base 2 The area of a trapezoid can be found the same way, but the formula becomes more complicated.

Area of a Trapezoid Base 1 height Base 2 The area of a trapezoid can be found the same way, but the formula becomes more complicated.

Area of a Trapezoid Base 1 height Base 2 The area of a trapezoid can be found the same way, but the formula becomes more complicated.

Area of a Trapezoid Base 1 height Base 2 The area of a trapezoid can be found the same way, but the formula becomes more complicated.

Area of a Trapezoid Base 1 height Base 2 The area of a trapezoid can be found the same way, but the formula becomes more complicated.

Area of a Trapezoid Base 1 height Base 2 The area of a trapezoid can be found the same way, but the formula becomes more complicated.

Area of a Trapezoid Base 1 height Base 2 The area of a trapezoid can be found the same way, but the formula becomes more complicated.

Area of a Trapezoid Base 1 height Base 2 The area of a trapezoid can be found the same way, but the formula becomes more complicated.

Area of a Trapezoid Base 1 height Base 2 The area of a trapezoid can be found the same way, but the formula becomes more complicated.

Area of a Trapezoid Base 1 Base 2 height Base 2 Base 1 The area of a trapezoid can be found the same way, but the formula becomes more complicated.

Area of a Trapezoid Base 1 Base 2 height Base 2 Base 1

This is the AVERAGE of the two bases! Area of a Trapezoid Base 1 Base 2 height Base 2 Base 1 Area of a trapezoid = This is the AVERAGE of the two bases!

Area of a Circle

Area of a Circle Circumference diameter radius

Area of a Circle In a circle, the Diameter is always equal Circumference In a circle, the Diameter is always equal to two times the radius d = 2r In a circle, the circumference diameter is always a little greater than 3. diameter radius This ratio of the circumference to the diameter of a circle Is called  and is approximately equal to 3.14

Area of a Circle

Area of a Circle

Area of a Circle

Area of a Circle

Area of a Circle

Area of a Circle

Area of a Circle

Area of a Circle

Area of a Circle

Area of a Circle

Area of a Circle

Area of a Circle

Area of a Circle

Area of a Circle

Area of a Circle

Area of a Circle

Area of a Circle

Area of a Circle

Area of a Circle

Area of a Circle

Area of a Circle

Area of a Circle

Area of a Circle

Area of a Circle

Area of a Circle

Area of a Circle radius

Area of a Circle radius This shows a circle broken up into ONLY 8 pieces. Suppose, we broke it up into a lot more!

Area of a Circle radius

Area of a Circle radius

Area of a Circle radius

Area of a Circle radius